Study on Set-Valued Inequalities based on Set-Valued Analysis and Convex Analysis and its Applications to Optimization Problems

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400194
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Niigata University
• Principal Investigator: TANAKA Tamaki 新潟大学, 自然科学系, 教授 (10207110)
• Co-Investigator(Renkei-kenkyūsha): TANINO Tetsuzo 大阪大学, 名誉教授 (50125605) ; MATSUSHITA Shinya 秋田県立大学, システム科学技術学部, 准教授 (20435449)
• Research Collaborator: SUZUKI Satoshi 島根大学, 総合理工学研究院, 助教 (70580489) ; KIMURA Kenji 芝浦工業大学, 工学部, 非常勤講師 ; HIGUCHI Masakazu 埼玉大学, 大学院理工学研究科, 産学官連携研究員
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 集合最適化 / ベクトル最適化 / 劣線形スカラー化 / 集合値写像 / スカラー化関数 / 集合値写像 / 二者択一の定理 / ファジィ集合

Outline of Final Research Achievements

In this study, we give several applications by using nonlinear scalarization methods for sets and proposed computational scheme for the scalarizing fucntions. This approach has been proposed in the previous study (research 21540121 supported by Grant-in-Aid for Scientific Research (C)). First, we clarified the inherited properties of the composite function of a monotonic scalarizing function and a sort of convex set-valued map systematically. As the results, we had several applications on certain generalization of Ricceri's nonlinear inequality to set-valued maps, alternative theorem for set-valued maps, and also we developed computational methods to evaluate the scalarizing functions and some construction of difference evaluation functions for fuzzy sets. Besides, we studied multicriteria two-persons zero-sum matrix games, algebraic properties of weak efficient solutions for sets, and hierarchical equilibrium problems including fixed-point problems.

Free Research Field

非線形解析学や凸解析学に基づいたベクトル最適化と集合最適化の研究